This monograph is part of a larger program, materializing in five volumes, whose principal aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings.
Volume II is concerned with function spaces measuring size and/or smoothness, such as Hardy spaces, Besov spaces, Triebel-Lizorkin spaces, Sobolev spaces, Morrey spaces, Morrey-Campanato spaces, spaces of functions of Bounded Mean Oscillations, etc., in general geometric settings. Work here also highlights the close interplay between differentiability properties of functions and singular integral operators. The text is intended for researchers, graduate students, and industry professionals interested in harmonic analysis, functional analysis, geometric measure theory, and function space theory.
Author(s): Dorina Mitrea, Irina Mitrea, Marius Mitrea
Series: Developments in Mathematics, 73
Edition: 1
Publisher: Springer
Year: 2023
Language: English
Pages: 949
City: Cham
Tags: Divergence Theorem; Parts Integration; Stokes Theorem; Singular Integral Operators; Function Spaces; Boundary Value Problems; Integral Representation Formulas In Complex Analysis
1
Prefacing the Full Series
Description of Volume II
Contents
978-3-031-13718-1_1
Preliminary Functional Analytic Matters
Algebraic and Operator Theoretic Rudiments
Quasi-Normed Spaces
Real Interpolation of Quasilinear Operators
Complex Interpolation on Quasi-Banach Spaces
Köthe Function Spaces and Other Categories of Topological Vector Spaces
978-3-031-13718-1_2
Abstract Fredholm Theory
Fredholm Theory in Banach Spaces
Fredholm Theory in Topological Vector Spaces
978-3-031-13718-1_3
Functions of Vanishing Mean Oscillations and Vanishing Hölder Moduli
Functions of Vanishing Mean Oscillations on Measure Metric Spaces
A New Brand of Hölder Spaces
978-3-031-13718-1_4
Hardy Spaces on Ahlfors Regular Sets
The Fefferman-Stein Grand Maximal Function
Lebesgue-Based and Lorentz-Based Hardy Spaces on Ahlfors Regular Sets
Real Interpolation of Hardy Spaces
Atomic Decompositions for Hardy Spaces
Molecules in Hardy Spaces
Duality in Hardy Spaces
Richness of the Duals of Lorentz-Based Hardy Spaces
Weak- Convergence and More on the Compatibility of Pairings
More on Hp Versus Lp: The Filtering Operator
978-3-031-13718-1_5
Banach Function Spaces, Extrapolation, and Orlicz Spaces
Generalized Banach Function Spaces
Extrapolation Theory
Young Functions and Orlicz Spaces
978-3-031-13718-1_6
Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals on Ahlfors Regular Sets
Morrey-Campanato Spaces and Their Pre-Duals on Ahlfors Regular Sets
Morrey Spaces and Their Pre-Duals on Ahlfors Regular Sets
978-3-031-13718-1_7
Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets
Definitions with Sharp Ranges of Indices and Basic Results
Atomic and Molecular Theory
Calderón's Reproducing Formula and Frame Theory
Interpolation of Besov and Triebel-Lizorkin Spaces via the Real Method
Complex Interpolation of Besov and Triebel-Lizorkin Spaces
Duality Results for Besov and Triebel-Lizorkin Spaces
Loose and Tight Embeddings
Envelopes of Besov and Triebel-Lizorkin Spaces
Intrinsic Characterizations of Besov Spaces
978-3-031-13718-1_8
Boundary Traces from Weighted Sobolev Spaces into Besov Spaces
Traces from Weighted Sobolev Spaces Defined in Rn
Technical Lemmas
Traces from Weighted Sobolev Spaces Defined in Open Subsets of Rn
Extension Operators from Boundary Besov Spaces into Weighted Sobolev Spaces
Weighted Sobolev Spaces of Negative Order, Duality, and the Conormal Derivative Operator
Traces from Weighted Maximal Sobolev Spaces in Open Subsets of Rn
978-3-031-13718-1_9
Besov and Triebel-Lizorkin Spaces in Open Sets
Besov and Triebel-Lizorkin Spaces in Rn
Besov and Triebel-Lizorkin Spaces in Open Subsets of Rn
Envelopes of Besov and Triebel-Lizorkin Spaces in Open Sets
Traces of Functions from Besov and Triebel-Lizorkin Spaces
The Conormal Derivative Operator on Besov and Triebel-Lizorkin Spaces
Extension from Boundary Besov Spaces into Smoothness Spaces in Open Sets
978-3-031-13718-1_10
Strong and Weak Normal Boundary Traces of Vector Fields in Hardy and Morrey Spaces
Strong Normal Boundary Traces of Vector Fields in Hardy Spaces
Weak Normal Boundary Traces of Vector Fields in Hardy and Morrey Spaces
The Manifold Setting
978-3-031-13718-1_11
Sobolev Spaces on the Geometric Measure Theoretic Boundary of Sets of Locally Finite Perimeter
Weak Tangential Derivatives and Boundary Sobolev Spaces
Distributional Tangential Derivatives
Distributional Derivatives Versus Weak Tangential Derivatives
The Tangential Gradient
Sobolev Spaces on Boundaries of Ahlfors Regular Domains …
Boundary Sobolev Spaces on Manifolds
A General Recipe for Sobolev-Like Spaces
Boundary Sobolev Spaces of Negative Smoothness
Principal-Value Distributions on Countably Rectifiable Sets
Hardy-Based Boundary Sobolev Spaces
Embedding Results Involving Boundary Sobolev Spaces
Real Interpolation Involving Boundary Sobolev Spaces
Morrey-Based, and Block-Based, Homogeneous Sobolev Spaces
1 (1)
Terms and Notation used in Volume II
References
